03 // Bidirectional STEM Calculators // MATHa_n = a₁ + (n - 1) × d

Arithmetic Sequence & N-th Term Calculator

An arithmetic progression advances by adding a constant common difference d to each preceding term, placing the n-th term at a_n = a₁ + (n - 1) × d.

a_n = a₁ + (n - 1) × d

Pick which value you want to calculate (or type into any field):Solving for: a_n — N-th Term Value

N-th Term ValueCalculated

Value of the sequence at position n

-100,000.00100,000.00

unit

First Term

Starting value at index n = 1

-100,000.00100,000.00

unit

Term Index Position

1-based index of the term

110,000.00

unit

Common Difference

Constant step added between consecutive terms

-1,000.001,000.00

unit

Calculated Result · N-th Term Value (a_n)

100

Step-by-Step Calculation

01Canonical Law: a_n = a₁ + (n - 1) × d

02Given Inputs (Base Units): a₁ = 5, n = 20, d = 5

03Solve for a_n (N-th Term Value): a_n = 100

An arithmetic progression advances by adding a constant common difference d to each preceding term, placing the n-th term at a_n = a₁ + (n - 1) × d.

Gauss Partial Sum (S_n)S_n = (n / 2) × (a₁ + a_n)

For default (a₁=5, a₂₀=100, n=20), S₂₀ = 1,050

Arithmetic Mean of Endpointsa_mid = (a₁ + a_n) / 2

Average of all terms equals average of first and last

Linear Function Equivalencef(n) = d·n + (a₁ - d)

Discrete linear slope d

N-th Term Value (a_n) · First Term (a₁)

Compare across:

Arithmetic Sequence & N-th Term Calculator — First Term → N-th Term Value
#	First Term (a₁)	N-th Term Value (a_n)	Δ %
01	-22,495.00	-22,400.00	-22500.00%
02	-19,863.42	-19,768.42	-19868.42%
03	-17,231.84	-17,136.84	-17236.84%
04	-14,600.26	-14,505.26	-14605.26%
05	-11,968.68	-11,873.68	-11973.68%
06	-9,337.11	-9,242.11	-9342.11%
07	-6,705.53	-6,610.53	-6710.53%
08	-4,073.95	-3,978.95	-4078.95%
09	-1,442.37	-1,347.37	-1447.37%
10	5 Current	100	0.00%
11	3,820.79	3,915.79	+3815.79%
12	6,452.37	6,547.37	+6447.37%
13	9,083.95	9,178.95	+9078.95%
14	11,715.53	11,810.53	+11710.53%
15	14,347.11	14,442.11	+14342.11%
16	16,978.68	17,073.68	+16973.68%
17	19,610.26	19,705.26	+19605.26%
18	22,241.84	22,336.84	+22236.84%
19	24,873.42	24,968.42	+24868.42%
20	27,505.00	27,600.00	+27500.00%

Was this tool helpful?

How do I find how many terms n are in a range from a₁ to a_n?+

Click SOLVE FOR on n (Term Index Position) to evaluate n = ((a_n - a₁) / d) + 1.

How do I calculate the sum of the first n terms S_n?+

Once a_n is solved, multiply the average of the first and last term by n: S_n = n × (a₁ + a_n) / 2.

Can the common difference d be negative?+

Yes. A negative d produces a strictly decreasing arithmetic progression.